Ab Ini'o Molecular Dynamics (MD) Simula?ons
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1 Ab Ini'o Molecular Dynamics (MD) Simula?ons Rick Remsing ICMS, CCDM, Temple University, Philadelphia, PA
2 What are Molecular Dynamics (MD) Simulations? Technique to compute statistical and transport properties of many-body systems Given initial condition (positions & momenta of N particles/ nuclei), evolve system in time following equation of motion
3 Why MD Simulations? We want to do statistical mechanics! Compute observables to connect with experiment Thermodynamics, structure, dynamics, electronic properties
4 Why MD Simulations? We want to do statistical mechanics! Compute observables to connect with experiment ho(r)i = R dro(r)e Z 3N-dimensional configurational integrals U(R) P hoi = n O ne Q Summation over all states " n Partition Function: Q = X n e " n H n = " n n
5 How to compute observables? Work under assumption of the Ergodic Hypothesis: Z hoi lim!1 0 dto(t)
6 How to compute observables? Work under assumption of the Ergodic Hypothesis: Z hoi lim!1 0 dto(t)
7 How to compute observables? Work under assumption of the Ergodic Hypothesis: Z hoi lim!1 0 dto(t) Long enough trajectories will adequately sample phase space
8 Simulating the Time Evolution Need to evaluate the forces on particles/nuclei Classical MD uses empirical potentials Numerous approaches exist for ab initio MD Ehrenfest dynamics Car-Parrinello MD (CPMD) Born-Oppenheimer MD (BOMD)
9 Born-Oppenheimer MD Work within the Born-Oppenheimer approximation Separation of nuclear & electronic DoFs Nuclei evolve in time Electronic structure problems is solution to time-independent Schrodinger equation: static problem BOMD equation of motion: m i r i (t) = r ri min 0 {h 0 H e 0i} H e 0 = E 0 0
10 Born-Oppenheimer MD Work within the Born-Oppenheimer approximation Separation of nuclear & electronic DoFs Nuclei evolve in time Electronic structure problems is solution to time-independent Schrodinger equation: static problem BOMD equation of motion: m i r i (t) = r ri min 0 {h 0 H e 0i} H e 0 = E 0 0 F
11 Integrating the equations of motion: the velocity Verlet algorithm Compute forces Update positions Update velocities Compute new Forces Update velocities Global error O( t 2 ) Need to carefully choose t
12 Exercise 1, Timesteps Run MD simulations of 8-atom liquid Si cell with t=0.5-3 fs Simulations in microcanonical (NVE) ensemble Energy is constant =0 Integration scheme should conserve energy, error in discrete integration will lead to buildup of errors
13 Results: Exercise 1, Timesteps Energy Fluctuations (and drift) increase with size of timestep Smaller t conserves E better Tradeoff between accuracy & efficiency
14 Exercise 2, Energy Convergence BOMD: Energy (and forces) depend on minimization of energy functional Sensitive to convergence criteria for self-consistent iteration Perform NVE BOMD for convergences of 1E-4 to 1E-7 ev for energy difference between iterations
15 Results: Exercise 2, Energy Convergence Energy drift increases significantly with decreasing tolerance Tradeoff between accuracy & efficiency
16 Exercise 3, Averages Compute average thermodynamic, structural, and dynamic observables Temperature in NVE ensemble Pair correlation functions Diffusion CoefFicient: Green-Kubo vs. Einstein relations
17 Results: Exercise 3a, Temperature <T> ~ 3000 K
18 Results: Exercise 3b, Structure Disordered liquid, ordered solid
19 Finite size effects in l-si
20 Exercise 3c, Dynamics MD also enables computation of dynamic properties Green-Kubo: Transport coefficients related to integral of time correlation function (TCF) Z 1 D = dt hv(0) v(t)i
21 Exercise 3c, Dynamics MD also enables computation of dynamic properties Green-Kubo: Transport coefficients related to integral of time correlation function (TCF) Z 1 D = dt hv(0) v(t)i Einstein: 6Dt = lim t!1 h r(t) r(0) i Should yield equivalent D values
22 Results: Exercise 3c, Dynamics from TCFs SCAN DSCAN ~ 0.4 Å 2 / ps DLDA ~ 1.4 Å 2 / ps LDA Green-Kubo type relations typically take longer to converge than MSD, for example
23 Results: Exercise 3d, Dynamics from MSD DSCAN ~ 0.76 Å 2 / ps SCAN DLDA ~ 1.35 Å 2 / ps Consistent with VACF results Faster dynamics in LDA LDA due to lack of good covalent bonds
24 Results: Exercise 3e, Phonon Density of States SCAN Phonon DOS is Fourier transform of velocity autocorrelation function, C(t) LDA Qualitatively similar Need better statistics Can compare with solid in phonon tutorial
25 Exercise 4, Canonical (NVT) Ensemble Perform simulations at constant temperature Use Andersen thermostat (preserves canonical distribution) Mimics coupling to heat bath (with a prescribed collision frequency) Particle momenta are reset randomly, following Maxwell- Boltzmann distribution Thermodynamics? Dynamics???
26 Results: Exercise 4, Canonical (NVT) Ensemble E is not the conserved quantity Can Fluctuate T Fluctuates about constant <T> Why isn t T constant? Small system, constant T as N tends to infinity
27 Results: Exercise 4, Canonical (NVT) Ensemble Structure is independent of ensemble
28 Results: Exercise 4, Canonical (NVT) Ensemble Dynamics are severely altered by thermostat Extreme case, others thermostats (and weak coupling) may not significantly alter dynamics Be careful!
29 SCAN description of silicon with MD
30 Background: Metallic and Covalent Bonding in Solid Silicon Si transitions from semiconducting diamond to metallic ß-Sn at P~12 GPa SCAN captures structure and energetics with quantitative accuracy Distinguishes J. Sun, et al. Nature Chem covalent from metallic bonds
31 Phase Transitions Coexistence: semiconducting solid and metallic liquid Tm ~ 1550 K, closer to Exp. (1680 K) than previous estimates using LDA: K RCR, J. Sun, M. L. Klein, to be submitted
32 Mixed Interactions: Liquid Silicon l-si is largely metallic; some covalent character 2nd peak is absent in LDA/ PBE descriptions LDA/PBE lack adequate discrimination between metallic and covalent bonding J. Sun, et al. Nature Chem. 2016
33 More phase transitions Thermodynamics Dynamics RCR, J. Sun, M. L. Klein, to be submitted
34 Changes in Electronic Structure NF DOS follows density dependence on T Transition from covalent and metallic bonding dominating RCR, J. Sun, M. L. Klein, to be submitted Covalent Heat Metallic
35 Q & A
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